392
9 Topologic Optimization of Vibrations of Size-Dependent Beams
the Poincaré map (9.71*e) exhibit the multi-frequency vibrations, and the points w(t)
and w(t + T ) are densely distributed, which indicates a chaotic structure.
Case study 2 (k 4 = 0.3, homogeneous beam). The forms of the signal (9.72a),
Fourier spectrum (9.72b), and particularly the wavelet spectrum (9.72c) imply chaotic
vibrations. The frequency Fourier spectrum exhibits ω p and ω 1 = 0.958, ω 2 =
3.791. It can be observed on the wavelet spectrum that ω 2 has time-dependent power.
The phase portrait (9.72d) and Poincaré map (9.72c) also indicate chaotic vibrations
of the beam.
Case study 2* (k 4 = 0.3, optimal microstructure). The signal (9.72*a) is periodic.
The Fourier spectrum (9.72*b) contains only the excitation frequency. The wavelet
spectrum (9.72*c) exhibits noisy frequencies localized in the low-frequency band.
The phase portrait (9.72*d) is qualitatively similar to the phase portrait (9.72*d) of
the previous case (θ =0). The Poincaré map (9.72*e) shows a point indicating the
existence of a periodic orbit.
Therefore, it is illustrated that in the case of absence of the temperature field, the
beam optimized material has an essential influence on the dynamic characteristics
of the beam.
In Table 9.10, the results for the temperature θ = −100C
◦ are shown, and they
are briefly described in what follows.
Case study 1 (k 4 = 0, homogeneous beam). The signal (9.71a) is chaotic. The
Fourier spectrum exhibits two of the most powerful frequencies ω 1 = 0.8947, ω 2 =
3.813 which satisfy the relations ω 1 = ω p /9.5 , ω 2 = ω p /2.223 . The phase portrait
(9.71d) and the Poincaré map (9.71c) also manifest chaotic beam vibrations.
Case study 1* (k 4 = 0, optimal microstructure). The signal (9.71*a) presents
a modified sinusoid-like vibration with the amplitude twice lower than in the case
of the homogeneous beam (9.71). The Fourier spectrum (9.71b) contains the frequencies ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 , ω 7 , ω 8 , ω 9 , ω 10 , and the excitation frequency ω p .
All frequencies are separated from each other by distances approximately equal to
0.775. The wavelet spectrum (9.71*c) supplements the Fourier spectrum with respect
to their time evolution. Namely, the frequencies ω 1 , ω 2 , ω 3 , ω 7 , ω 10 appear in the
spectrum beginning with the time instant t ≈ 850. Points of the Poincaré maps are
located on a ring, implying regular vibrations.
Case study 2 (k 4 = 0.3, homogeneous beam). The forms of the signal (9.72a),
Fourier spectrum (9.72b), and particularly the Morlet spectrum (9.72c) indicate
chaotic vibrations. The spectrum, besides ω p , exhibits two dominating frequencies ω 1 = 0.9604, ω 2 = 3.777, which satisfy the estimation ω 1 = ω p /8.85 , ω 2 =
ω p /2.25 . The wavelet spectrum illustrates how ω 2 changes its power over time in
a way analogous to the case 1. The phase portrait (9.72d) and Poincaré map (9.72c)
also indicate beam chaotic vibrations.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
the Poincaré map (9.71*e) exhibit the multi-frequency vibrations, and the points w(t)
and w(t + T ) are densely distributed, which indicates a chaotic structure.
Case study 2 (k 4 = 0.3, homogeneous beam). The forms of the signal (9.72a),
Fourier spectrum (9.72b), and particularly the wavelet spectrum (9.72c) imply chaotic
vibrations. The frequency Fourier spectrum exhibits ω p and ω 1 = 0.958, ω 2 =
3.791. It can be observed on the wavelet spectrum that ω 2 has time-dependent power.
The phase portrait (9.72d) and Poincaré map (9.72c) also indicate chaotic vibrations
of the beam.
Case study 2* (k 4 = 0.3, optimal microstructure). The signal (9.72*a) is periodic.
The Fourier spectrum (9.72*b) contains only the excitation frequency. The wavelet
spectrum (9.72*c) exhibits noisy frequencies localized in the low-frequency band.
The phase portrait (9.72*d) is qualitatively similar to the phase portrait (9.72*d) of
the previous case (θ =0). The Poincaré map (9.72*e) shows a point indicating the
existence of a periodic orbit.
Therefore, it is illustrated that in the case of absence of the temperature field, the
beam optimized material has an essential influence on the dynamic characteristics
of the beam.
In Table 9.10, the results for the temperature θ = −100C
◦ are shown, and they
are briefly described in what follows.
Case study 1 (k 4 = 0, homogeneous beam). The signal (9.71a) is chaotic. The
Fourier spectrum exhibits two of the most powerful frequencies ω 1 = 0.8947, ω 2 =
3.813 which satisfy the relations ω 1 = ω p /9.5 , ω 2 = ω p /2.223 . The phase portrait
(9.71d) and the Poincaré map (9.71c) also manifest chaotic beam vibrations.
Case study 1* (k 4 = 0, optimal microstructure). The signal (9.71*a) presents
a modified sinusoid-like vibration with the amplitude twice lower than in the case
of the homogeneous beam (9.71). The Fourier spectrum (9.71b) contains the frequencies ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , ω 6 , ω 7 , ω 8 , ω 9 , ω 10 , and the excitation frequency ω p .
All frequencies are separated from each other by distances approximately equal to
0.775. The wavelet spectrum (9.71*c) supplements the Fourier spectrum with respect
to their time evolution. Namely, the frequencies ω 1 , ω 2 , ω 3 , ω 7 , ω 10 appear in the
spectrum beginning with the time instant t ≈ 850. Points of the Poincaré maps are
located on a ring, implying regular vibrations.
Case study 2 (k 4 = 0.3, homogeneous beam). The forms of the signal (9.72a),
Fourier spectrum (9.72b), and particularly the Morlet spectrum (9.72c) indicate
chaotic vibrations. The spectrum, besides ω p , exhibits two dominating frequencies ω 1 = 0.9604, ω 2 = 3.777, which satisfy the estimation ω 1 = ω p /8.85 , ω 2 =
ω p /2.25 . The wavelet spectrum illustrates how ω 2 changes its power over time in
a way analogous to the case 1. The phase portrait (9.72d) and Poincaré map (9.72c)
also indicate beam chaotic vibrations.